What are the implications of Gödel's incompleteness theorems for the foundations of mathematics, and how do they impact our understanding of mathematical truth and consistency?

Gödel’s incompleteness theorems demonstrate that any sufficiently rich formal system of mathematics contains true statements that cannot be proven within that system, as well as statements that can neither be proven nor disproven. This challenges the idea of a complete and consistent mathematical theory and has profound implications for the philosophy of mathematics.